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Affine balance: algebraic spacetime functionalism as a guide to identifying spacetime

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Our two most empirically successful theories, quantum mechanics and general relativity, are at odds with each other when it comes to several foundational issues. The deepest of these issues is also, perhaps, the easiest to grasp intuitively: what is spacetime? Most attempts at theories of quantum gravity do not make it obvious which degrees of freedom are spatiotemporal. In non-general relativistic theories, the matter/spacetime distinction is adequately tracked by the dynamical/non-dynamical object distinction. General relativity is different, because spacetime, if taken to be jointly, but with some redundancy, represented by a smooth manifold and a metric tensor field, is not an immutable, inert, external spectator. Our dynamical/non-dynamical distinction appears no longer to do the work for us; we appear to need something else. In the first part of this talk, I push back against the idea that the dynamical/non-dynamical distinction is doomed. I motivate a more general algebraic characterisation of spacetime based on Eleanor Knox’s spacetime functionalism, and the Helmholtzian notion of free mobility. I argue that spacetime is most usefully characterised by its (local) affine structure.

In the second part of this talk, I consider the debate between Harvey Brown and Oliver Pooley, on one hand, and Michel Janssen and Yuri Balashov, on the other, about the direction of the arrow of explanation in special relativity. Characterising spacetime using algebraic functionalism, I demonstrate that only Brown’s position is neutral on the substantivalism–relationalism debate. This neutrality may prove to be highly desirable in an interpretation of spacetime that one hopes will generalise to theories of quantum gravity—it seems like poor practice to impose restrictions on an acceptable quantum theory of spacetime based on metaphysical prejudices or approximately true effective field theories. The flexibility of Brown’s approach affords us a theory-dependent a posteriori identification of spacetime, and arguably counts in its favour. I conclude by gesturing towards how this construction might be useful in extending Brown’s view to supersymmetric field theories (and theories of quantum gravity).

This talk is part of the CamPoS (Cambridge Philosophy of Science) seminar series.

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